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Mirrors > Home > MPE Home > Th. List > raleqbidvv | Structured version Visualization version GIF version |
Description: Version of raleqbidv 3303 with additional disjoint variable conditions, not requiring ax-8 2114 nor df-clel 2809. (Contributed by BJ, 22-Sep-2024.) |
Ref | Expression |
---|---|
raleqbidvv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
raleqbidvv.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
raleqbidvv | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqbidvv.2 | . . . . . 6 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | alrimiv 1935 | . . . . 5 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
3 | raleqbidvv.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | dfcleq 2729 | . . . . . 6 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
5 | 3, 4 | sylib 221 | . . . . 5 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
6 | 19.26 1878 | . . . . 5 ⊢ (∀𝑥((𝜓 ↔ 𝜒) ∧ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) ↔ (∀𝑥(𝜓 ↔ 𝜒) ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) | |
7 | 2, 5, 6 | sylanbrc 586 | . . . 4 ⊢ (𝜑 → ∀𝑥((𝜓 ↔ 𝜒) ∧ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) |
8 | imbi12 350 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ((𝜓 ↔ 𝜒) → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)))) | |
9 | 8 | impcom 411 | . . . 4 ⊢ (((𝜓 ↔ 𝜒) ∧ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒))) |
10 | 7, 9 | sylg 1830 | . . 3 ⊢ (𝜑 → ∀𝑥((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒))) |
11 | albi 1826 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))) | |
12 | 10, 11 | syl 17 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))) |
13 | df-ral 3056 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
14 | df-ral 3056 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)) | |
15 | 12, 13, 14 | 3bitr4g 317 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ∈ wcel 2112 ∀wral 3051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-cleq 2728 df-ral 3056 |
This theorem is referenced by: rexeqbidvv 3306 raleqbi1dv 3307 postcposALT 45976 |
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